Support vector machines (SVM) are a powerful tool to analyze data with a number of predictors approximately equal or larger than the number of observations. However, originally, application of SVM to analyze biomedical data was limited because SVM was not designed to evaluate importance of predictor variables. Creating predictor models based on only the most relevant variables is essential in biomedical research. Currently, substantial work has been done to allow assessment of variable importance in SVM models but this work has focused on SVM implemented with linear kernels. The power of SVM as a prediction model is associated with the flexibility generated by use of non-linear kernels. Moreover, SVM has been extended to model survival outcomes. This paper extends the Recursive Feature Elimination (RFE) algorithm by proposing three approaches to rank variables based on non-linear SVM and SVM for survival analysis. The proposed algorithms allows visualization of each one the RFE iterations, and hence, identification of the most relevant predictors of the response variable. Using simulation studies based on time-to-event outcomes and three real datasets, we evaluate the three methods, based on pseudo-samples and kernel principal component analysis, and compare them with the original SVM-RFE algorithm for non-linear kernels. The three algorithms we proposed performed generally better than the gold standard RFE for non-linear kernels, when comparing the truly most relevant variables with the variable ranks produced by each algorithm in simulation studies. Generally, the RFE-pseudo-samples outperformed the other three methods, even when variables were assumed to be correlated in all tested scenarios. The proposed approaches can be implemented with accuracy to select variables and assess direction and strength of associations in analysis of biomedical data using SVM for categorical or time-to-event responses. Conducting variable selection and interpreting direction and strength of associations between predictors and outcomes with the proposed approaches, particularly with the RFE-pseudo-samples approach can be implemented with accuracy when analyzing biomedical data. These approaches, perform better than the classical RFE of Guyon for realistic scenarios about the structure of biomedical data.

Comparisons of recent with historical samples of chromosome inversion frequencies provide opportunities to determine whether genetic change is tracking climate change in natural populations. We determined the magnitude and direction of shifts over time (24 years between samples on average) in chromosome inversion frequencies and in ambient temperature for populations of the fly Drosophila subobscura on three continents. In 22 of 26 populations, climates warmed over the intervals, and genotypes characteristic of low latitudes (warm climates) increased in frequency in 21 of those 22 populations. Thus, genetic change in this fly is tracking climate warming and is doing so globally.

The parametric statistical models with suitable regularity conditions have a natural Riemannian manifold structure, given by the information metric. Since the parameters are merely labels for the probability measures, an inferential statement should be formulated through intrinsic objects, invariant under reparametrizations. In this context the estimators will be random objects valued on the manifold corresponding to the statistical model. In spite of these considerations, classical measures of an estimator's performance, like the bias and the mean square error, are clearly dependent on the statistical model parametrizations. In this paper the authors work with extended notions of mean value and moments of random objects which take values on a Hausdorff and connected manifold, equipped with an affine connection. In particular, the Riemannian manifold case is considered. This extension is applied to the bias and the mean square error study in statistical point estimation theory. Under this approach an intrinsic version of the Cramer-Rao lower bound is obtained: a lower bound, which depends on the intrinsic bias and the curvature of the statistical model, for the mean square of the Rao distance, the invariant measure analogous to the mean square error. Further, the behavior of the mean square of the Rao distance of an estimator when conditioning with respect to a sufficient statistic is considered, obtaining intrinsic versions of the Rao-Blackwell and Lehmann-Scheffe theorems. Asymptotic properties complete the study.

Article AN EXPLICIT SOLUTION OF INFORMATION GEODESIC EQUATIONS FOR THE MULTIVARIATE NORMAL MODEL was published on February 1, 1991 in the journal Statistics & Risk Modeling (volume 9, issue 1-2).